Dirac notation

February 2, 2025

Dirac notation is a compact language for quantum states and operators.

Kets, bras, and inner products

A state vector is written as a ket:

\lvert \psi \rangle.

If you choose a basis $\{\lvert 0 \rangle, \lvert 1 \rangle\}$ , you can expand a qubit state as

\lvert \psi \rangle = \alpha \lvert 0 \rangle + \beta \lvert 1 \rangle,

where $\alpha,\beta \in \mathbb{C}$ are complex amplitudes.

The conjugate-transpose (Hermitian adjoint) of a ket is a bra:

\langle \psi \rvert.

In matrix terms:

$\lvert \psi \rangle$ behaves like a column vector
$\langle \psi \rvert$ behaves like a row vector

Multiplying a bra by a ket gives an inner product:

\langle \phi \vert \psi \rangle.

You will often see $\langle \phi \vert \psi \rangle$ written with the middle bars “merged” into a single bar; this is where the word bra-ket comes from.

Inner products in quantum mechanics use complex conjugation. For example,

\langle \psi \vert \psi \rangle

is always a non‑negative real number.

Operators acting on states

Linear maps (matrices) are written as operators, such as $A$ . Applying an operator to a ket is written as

A\lvert \psi \rangle.

This is simply “matrix times vector,” producing a new ket.

It is also common to take an inner product after applying an operator:

\langle \phi \vert A \vert \psi \rangle.

Read this as: start with $\lvert \psi \rangle$ , apply $A$ , then take the inner product with $\lvert \phi \rangle$ .

Expectation values (a common pattern)

When $A$ is an observable (Hermitian operator), the expected value in state $\lvert \psi \rangle$ is

\langle A \rangle = \langle \psi \vert A \vert \psi \rangle.

This expression shows up everywhere in quantum computing: energies, measurement statistics, and algorithm analysis.

Basis states, wavefunctions, and “resolution of the identity”

One common source of confusion is that the same $\langle \cdot \vert \cdot \rangle$ notation is used both for:

the inner product between two abstract states, and
the components of a state in a particular basis.

Discrete basis (finite or countable)

If $\{\lvert i \rangle\}$ is an orthonormal basis, then the coefficients

\psi_i \equiv \langle i \vert \psi \rangle

are just the components of $\lvert \psi \rangle$ in that basis, and you can reconstruct the state as

\lvert \psi \rangle = \sum_i \lvert i \rangle \langle i \vert \psi \rangle.

This motivates the identity operator:

I = \sum_i \lvert i \rangle \langle i \rvert.

Continuous basis (position as an example)

For position, the basis is labeled by a continuous variable $x$ (or $\mathbf{r}$ in 3D). The wavefunction is defined by

\psi(x) \equiv \langle x \vert \psi \rangle,

and its complex conjugate is

\psi^*(x) = \langle \psi \vert x \rangle.

The “sum over basis states” becomes an integral:

\lvert \psi \rangle = \int dx\, \lvert x \rangle \langle x \vert \psi \rangle = \int dx\, \psi(x)\lvert x \rangle.

Accordingly, the identity operator is written as

I = \int dx\, \lvert x \rangle \langle x \rvert.

This is called a resolution of the identity. It means: if you insert this operator anywhere, it does nothing—but it lets you rewrite abstract expressions in a concrete basis.

The objects

\lvert x \rangle

are “generalized basis vectors.” The integral formulas are shorthand that work well in physics, but they require some mathematical care (distributions) in a rigorous treatment.

Why operators are not “just chilling” in $\langle \psi \vert A \vert \psi \rangle$

The operator $A$ acts on the ket $\lvert \psi \rangle$ first, producing a new state $A\lvert \psi \rangle$ . Then the bra $\langle \psi \rvert$ takes an inner product with that new state.

If you insert a resolution of the identity in the position basis, you get a more familiar-looking integral:

\langle \psi \vert A \vert \psi \rangle = \int dx\, \langle \psi \vert x \rangle \langle x \vert A \vert \psi \rangle.

When $A$ is diagonal in the $\lvert x \rangle$ basis (for example, the position operator $\hat{x}$ ), this reduces to the classical-looking form:

\langle A \rangle = \int dx\, \psi^*(x)\, A(x)\, \psi(x).

So the operator is absolutely doing work—it is what connects the abstract state to the measurable quantity you care about.

Tensor products

Multi-qubit kets use the tensor product symbol (often omitted in writing):

\lvert 0 \rangle \otimes \lvert 1 \rangle = \lvert 01 \rangle.

Operators on separate qubits combine as Kronecker products $A \otimes B$ .

For a practical guide (including partial trace and reduced density matrices), see Tensor products and partial trace.

If phases still feel “mysterious”, learn the core trick used by many algorithms: Phase kickback.

Projectors and measurement

A projective measurement can be described using projectors onto orthogonal subspaces. For example, the projector onto $\lvert 0 \rangle$ on a single qubit is $P_0 = \lvert 0 \rangle \langle 0 \rvert$ .

When you read circuit diagrams, translate each time slice into the corresponding unitary acting on the tensor-product Hilbert space.

Matrices as gates

Linear algebra refresher Matrices as gates